how many of each type should be made in order to obtain the maximum possible profits if the profit on each spots cycle is $1 and on each racing model is $3

Question

A bicycle manufacturer makes two models a sports cycle and a racing machine. In order to make a sports model, 6 man hours are needed, while a racing model requires 10 man hours. (By 6 man hours we mean either 1 man working for 6 hours, or 2 men working for 3 hours or 6 men working for 1 hour etc). the manufacturer can employ no more than 15 men and these men work for 8 hours per day for 5 days each week. the cost of materials amounts to $5 per cycle and manufacturer's total weekly quota of such materials may not exceed $400, the firm has a contact to supply at least 30 sports model and 20 racers per week. how many of each type should be made in order to obtain the maximum possible profits if the profit on each spots cycle is $1 and on each racing model is $3

Answer 1

Profit function is Profit(x,y) = x + 3y

 

Man hour constraints:

   6x+10y <= 600 <--- 3x + 5y < 300    <<--- 15 employees  x 40 hours per week = 600 man hours available

 

Material and Capital constraints:

 5x + 5y <= 400  ---> x + y <= 80

 

 

x>=30  y>=20

 

THe vertices of the critical region:

(30,42) --> $156 profit

(50,30) --> $140 profit

(60,20) --> $120 profit

(30,20) ---> $90 profit

 

30 and 42